First, we show that the lifting operator \({\mathcal {J}}\)  given by
\({\mathcal {J}}f = {\mathcal {F}}^{-1} (1+(\cdot )^2)^{1/2} {\mathcal {F}}f \quad (f\in {\mathcal {S}}(X))\) 
maps \(B^{s,q}_\Phi (X)\)  isomorphically onto \(B^{s-1,q}_{\Phi }(X)\) .
Let \(\lbrace \psi _j\rbrace _{j\in N_{0}}\)  be the resolution of the identity on \(\mathbb {R}\)  as in Section . Set
\(\phi _j(t):=2^{-j}(1+t^2)^{1/2} \psi _j(t) \quad (j\in \mathbb {N}_0).\) 
In order to show that \({\mathcal {J}}f\in B^{s-1,q}_\Phi (X)\)  for every \(f\in B^{s,q}_\Phi (X)\)  it is sufficient to show that there exists a constant \(\mu \)  such that for all \(j\in \mathbb {N}_0\)
\(\left\Vert  2^{(s-1)j} \psi _j(D) {\mathcal {J}}f \right\Vert _\Phi (X) \le \mu \left\Vert  2^{sj} \psi _j(D)f \right\Vert _\Phi (X).\) 
For note that for every \(f\in {\mathcal {S}}^{\prime }(X)\)  and every \(t\in \mathbb {R}\)
\(| 2^{(s-1)j} \psi _j(D) {\mathcal {J}}f (t) |_X & = | 2^{js} \phi _j (D) f(t) |_X \\& \le 2^{js} \sup _{r\in \mathbb {R}} \frac{|\phi _j(D) f (t - r) |_X}{1+ |2^{j} r|} =: 2^{sj} \phi ^\ast _j f(t).\) 

We show that there exists a constant \(\mu \)  such that for every \(f\in B^{s,q}_\Phi (X)\)  and every \(j\in \mathbb {N}_0\)  we have that
\(\Vert  \phi ^\ast _jf\Vert _\Phi \le \mu \Vert  \phi _j(D) f\Vert _{\Phi (X)} \quad \textrm { and } \quad \Vert  \phi _j(D) f\Vert _{\Phi (X)} \le \mu \Vert  \psi _j(D) f\Vert _{\Phi (X)}.\) 
The second inequality above follows from Proposition REF  since
\(\phi _j(D) f = {\mathcal {F}}^{-1} 2^{-j} (1+(\cdot )^2)^{1/2} \chi _j {\mathcal {F}}{\mathcal {F}}^{-1} \psi _j {\mathcal {F}}f\) 
and, as it is readily seen, the functions \(\rho _j: \mathbb {R}\ni t \mapsto 2^{-j} (1+t^2)^{1/2} \chi _j(t)\) , \(j\in \mathbb {N}_0\) , satisfy the \(\widetilde{\mathfrak {M}}_3\) -condition uniformly in \(j\) . Therefore,
\(\rho _j(D)\) , \(j\in N_0\) , restrict to uniformly bounded operators in \({\mathcal {L}}(B^{s,q}_\Phi (X))\) .
To get the first inequality in (REF ), by the same argument as in the proof of [1]}, we obtain the existence of a constant \(\mu \)  such that for every \(g\in {\mathcal {S}}(X)\)  and \(j\in N_0\)
\(\phi ^\ast _j g(t) \le \mu M\left(|\phi _j(D)g|_X \right)(t).\) 
Now by an approximation argument similar to that given in the proof of [1]} one can show that for every Muckenhoupt weight \(w\in A_2\)  for every \(f\in L^2_w(X)\)  we have that
\(\Vert \phi ^\ast _j f\Vert _{L^2_w} \le \mu \Vert  M\Vert _{L^2_w} \Vert  \phi _j(D) f\Vert _{L^2_w(X)}.\) 
Since \(\Vert M\Vert _{L^2_w}\lesssim [w]^2_{A_2}\)  (see [3]}), by an argument similar to that from the proof of Lemma REF (the case \(\mathbb {E}= F^{s,q}_\Phi \) ), we get the first inequality in (REF ).
Therefore, \({\mathcal {J}}\)  maps \(B^{s,q}_\Phi (X)\)  into \(B^{s-1,q}_\Phi (X)\) .
Note that \({\mathcal {J}}\)  is an isomorphism onto \({\mathcal {S}}^{\prime }(X)\) , and by the same argument to that above one can show that \({\mathcal {J}}^{-1} f = {\mathcal {F}}^{-1}(1+(\cdot )^2)^{-1/2}{\mathcal {F}}f \in B^{s,q}_\Phi (X)\)  for each \(f \in B^{s-1,q}_\Phi (X)\) . Therefore, the lifting property of \({\mathcal {J}}\)  is proved.
Now fix \(f\in B^{s,q}_\Phi (X)\) . We show that \(f, \partial f\in B^{s-1,q}_\Phi (X)\) .
Obviously, \(f\in B^{s-1,q}_\Phi (X)\) . For \(\partial f\) , note first that the function \(\rho (t) := t(1+t^2)^{-1/2}\) , \(t\in \mathbb {R}\) , satisfies the \(\widetilde{\mathfrak {M}}_3\) -condition and is of \({\mathcal {O}}_M\) -class. Thus, by Proposition REF , \(\rho (D)\)  restricts to an operator in \({\mathcal {L}}(B^{s-1,q}_\Phi (X))\) , and
\(\Vert  \partial f \Vert _{B^{s-1,q}_\Phi (X)} = \Vert \rho (D) {\mathcal {J}}f\Vert _{B^{s-1,q}_\Phi (X)} \le \Vert  \rho (D) {\mathcal {J}}\Vert _{{\mathcal {L}}(B^{s-1,q}_\Phi (X))} \Vert f\Vert _{{B^{s,q}_\Phi (X)}}\) 
Finally, let \(\eta (t) := ((1+t^2)^{1/2} - 1)t^{-1}\) . It is readily seen that \(\eta \)  satisfies the \(\widetilde{\mathfrak {M}}_3\) -condition. Therefore, \(\eta (D)\)  restricts to an operator in \({\mathcal {L}}(B^{s,q}_\Phi (X))\)  and for every \(f\in {\mathcal {S}}^{\prime }(X)\)
\({\mathcal {J}}f = f + \eta (D) \partial f.\) 
Since \({\mathcal {J}}^{-1}\)  maps isomorphically \({B}^{s-1,q}_\Phi (X)\)  onto \({B}^{s,q}_\Phi (X)\) , if \(f \in B^{s-1,q}_\Phi (X)\)  with \(\partial f\in B^{s-1,q}_\Phi (X)\) , then \(f \in B^{s,q}_\Phi (X)\) . It completes the proof.
